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Full text of "Treatise On Analysis Vol-Ii"

1    EXISTENCE AND UNIQUENESS OF HAAR MEASURE        247

Now extend I to the whole of Jf R(G) by putting 1(0) = 0 and, for each
/=/i -/2 with/1?/2 in Jf*, !(/) = I(/i) - I(/2). From (14.1.5.3) it follows
immediately that !(/) depends only on/and not on the choice of expression
/=/i ~/2- Clearly (14.1.53) remains valid for all /;/' in jfR(G). Further-
more, the original definition of I shows immediately that I(A/) = AI(/) for all
/e Jf * and all real numbers A > 0; and this relation obviously extends to the
case where fe JfR(G) and AeR. We may therefore conclude from (13.3.1)
that I is a positive measure on G; also I is not zero and, by construction,
I(y($)/) = !(/) for all fe JfR(G). In other words, we have constructed a
nonzero left-invariant positive measure on G.

(2) Uniqueness. Let \L (resp. v) be a nonzero left (resp. right) invariant
measure on G. Then v is a left-invariant measure. We shall show that p. and v
are proportional, and this will complete the proof of (14.1.5). Let/e 2TC(G)
be such that n(f) ^ 0, and consider the function Df on G defined by

I

f(rls)dv(f).

We shall show that D^ is continuous on G. In fact this will be a consequence of
the following more general result :

(1 4.1 .5.5) Let Gbea locally compact group, H a closed subgroup, a a measure
on H, and f: G-C a continuous mapping. Suppose that either Supp(/) is
compact or Supp(a) is compact. Then the mappings

f(si) da(0       and       sh->

f(ts) d
are continuous on G.

Consider, for example, the first of these integrals. Let ^0 e G and let V0 be
a compact neighborhood of s0 . Given e > 0, we have to find a neighborhood

V cV0 of Jo such that for all s e V we have.|J(/(tf) -f(s0 1)) dot(t) ^ e. If
K = Supp(/) is compact and L = \VJK, then

- /(s0 0) rfa(0;

since / is uniformly continuous (with respect to a right-invariant distance
on G) (3.16,5), there exists a neighborhood W of e in G such that the
relation s e W5-0 implies \f(st) f(s0t)\ rg e/||(L) for all t E G, and we may
take V = V0 n W^0 . If on the other hand S = Supp(a) is compact, we havebecause/i-^In( |/j) is a seminorm.